point-group assignment flowchart

Faced with a molecule, how do you decide whether it is C2v, D3h, or Td without memorizing dozens of cases? You run it through a short decision tree, asking a fixed sequence of yes-or-no questions about which symmetry elements it has. Each answer prunes the possibilities until only one point group is left. The flowchart is the practical recipe that turns the abstract idea of symmetry into a routine you can do in under a minute.

The standard route goes roughly like this. First, special cases: is it linear? (If yes, it is C-infinity-v or D-infinity-h depending on whether it has a center of inversion.) Is it one of the high-symmetry polyhedral shapes — tetrahedral Td, octahedral Oh, icosahedral Ih? If not, find the principal axis (the highest-order Cn). Then ask: are there n twofold axes perpendicular to it? If yes you are in the D family; if no you are in the C (or S) family. Finally, look at mirror planes: a horizontal plane (sigma-h) gives the "h" subscript (C2h, D3h), vertical/dihedral planes give the "v" or "d" subscript (C2v, D3d), and no planes leave you with plain Cn or Dn (or Sn if only an improper axis survives).

Learning to walk the flowchart reliably is the gateway skill of this whole subject, because every later payoff — reading a character table, predicting spectra, building symmetry-adapted orbitals — starts by knowing the point group. The most common slips are forgetting to check for the center of inversion in linear molecules, missing hidden Sn axes, and choosing the wrong principal axis. Done carefully and in order, though, the chart is foolproof: the questions are designed so the answers can only lead to one destination.